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For the second question: The implied density is the density function we integrate call payoffs against to match market call prices, denoted $f$ here. So, ignoring discount factors, the answer comes from Dirac delta function's properties: $$\mathbf{E}[\delta(S-K)] = \int_{-\infty}^\infty \delta(S-K)f(S) dS = f(K)$$ Alternatively:  C = C(K) = \int_K^\infty ...

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Does the integrated CIR process itself by any chance has such a representation in the form of the sum of independent random variables? I think the answer to this is clearly "no." The CIR process is (as @DavidAddison points out in comments above) like an Ornstein-Uhlenbeck process. The mean-reverting property of the O-U (and the CIR) mean that the ...

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If ${\frac {\partial^2 C} {\partial K ^2}}$ was zero, then the price-strike curve would just be a straight sloping-downwards line, and it would cost the same to buy either two call options at strike $K$ (portfolio A), or one option each at strike $K-1$ and strike $K+1$ (portfolio B). If you think about the payoffs at expiry where spot=$S_t$ of these two ...

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Here is a snip that will create and plot a Heston vol surface import numpy as np import QuantLib as ql from matplotlib import pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Utility function to plot vol surfaces (can pass in ql.BlackVarianceSurface objects too) def plot_vol_surface(vol_surface, plot_years=np.arange(0.1, 2, 0.1), plot_strikes=np....

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Usually multidimensional objective function of calibration error of stochastic volatility models (Heston , bergomi etc) have many local minima, thus you would get similar calibration error for very different set of parameters. Some ways to deal with it: specify paramter range your are comfortable with. let's say you want your vol of vol to be in the region ...

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The stochastic volatility model is calibrated to (a subset of) vanilla option prices. When the implied volatility is shifted to calculate vega, the model is calibrated again. Although pure stochastic volatility models can only match a few vanilla prices, a local volatility component is usually included nowadays. This allows the calibration to hit the whole ...

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